Vilmos Totik

Logarithmic potentials with external fields

This treatment of potential theory emphasizes the effects of an external field (or weight) on the minimum energy problem. Several important aspects of the external field problem (and its extension to signed measures) justify its special attention. The most striking is that it provides a unified approach to seemingly different problems in constructive analysis. These include the asymptotic analysis of orthogonal polynomials, the limited behavior of weighted Fekete points; the existence and construction of fast decreasing polynomials; the numerical conformal mapping of simply and doubly connected domains; generalization of the Weierstrass approximation theorem to varying weights; and the determination of convergence rates for best approximating rational functions.

General Orthogonal Polynomials by Herbert Stahl

Introduction 1. Upper and lower bounds 2. Zero distribution of orthogonal polynomials 3. Regular n-th root asymptotic behaviour of orthogonal polynomials 4. Regularity criteria 5. Localization 6. Applications Appendix Notes and bibliographical references Bibliography List of symbols Index.

Polynomial inverse images and polynomial inequalities

valid for polynomials Pn of degree at most n. In this paper we are primarily interested in what form these inequalities take on several intervals. We shall see that the extension to general sets involves the equilibrium measure of these sets. We shall give the precise form of the Bernstein inequality for arbi trary compacts, and an asymptotical ly best form of the Markoff inequality for sets consisting of finitely many intervals. Actually, in this case we shall prove different Markoff inequalities one-one-associated with each one of the endpoints of the system of intervals. The proofs will heavily use sets that are obtained as the inverse images of intervals under (special) polynomial mappings. We shall see that the original Bernstein and

Asymptotics for Christoffel functions for general measures on the real line

We consider asymptotics of Christoffel functions for measures ν with compact support on the real line. It is shown that under some natural conditionsn times thenth Christoffel function has a limit asn→∞ almost everywhere on the support, and the limit is the Radon-Nikodym derivative of ν with respect to the equilibrium measure of the support of ν. The case in which the support is an interval was settled previously by A. Máté, P. Nevai and the author. The present paper solves the general problem.

Szego5's extremum problem on the unit circle

It is shown that the Christoffel functions arising from the Szegd extremum problem associated with a finite positive Borel measure on the interval [- wr, wr) satisfy

Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle

Consider a system {φn} of polynomials orthonormal on the unit circle with respect to a measuredμ, withμ′>0 almost everywhere. Denoting bykn the leading coefficient ofφn, a simple new proof is given for E. A. Rakhmanovs important result that limn→∞,kn/kn+1=1; this result plays a crucial role in extending Szegös theory about polynomials orthogonal with respect to measuresdμ with logμ′∈L1 to a wider class of orthogonal polynomials.

Weighted polynomial inequalities

For the weights exp (−|x|λ), 0<λ≤1, we prove the exact analogue of the Markov-Bernstein inequality. The Markov-Bernstein constant turns out to be of order logn for λ=1 and of order 1 for 0<λ<1. The proof is based on the solution of the problem of how fast a polynomialPn can decrease on [−1,1] ifPn (0)=1. The answer to this problem has several other consequences in different directions; among others, it leads to a general theorem about the incompleteness of the set of polynomials in weightedLp spaces.

Fast decreasing polynomials

Matching two-sided estimates are given for the minimal degree of polynomialsP satisfyingP(0)=1 and ¦P(x)|≤exp(−ϕ (¦x¦)),x ∈ [−1,1], whereϕ is an arbitrary, in [0, 1], increasing function. Besides these fast decreasing polynomials we also consider bell-shaped polynomials and polynomials approximating well the signum function.

Necessary conditions for weighted mean convergence of Fourier series in orthogonal polynomials

Abstract Necessary conditions are found for weighted mean convergence of Fourier series in orthogonal polynomials corresponding to measures dα with support [−1, 1] for which α′ > 0 almost everywhere in [−1, 1]. Some additional properties of such orthogonal polynomials are also proved.

Online algorithms for a dual version of bin packing

Abstract It is shown that for the dual version of bin packing defined by Assmann et al. no online algorithm can have a performance ratio better than 1 2 . For uniformly distributed elements we give an asymptotically optimal algorithm.